MathDB

Problem 7

Part of 2023 VN Math Olympiad For High School Students

Problems(2)

prove that AB=AC in each case

Source: 2023 Viet Nam math olympiad for high school students D1 P7

3/25/2023
Given a triangle ABCABC with symmedians BE,CF(E,FBE,CF(E,F are on the sides CA,AB,CA,AB, respectively)) intersecting at Lemoine point L.L. Prove that: AB=ACAB=AC in each case: a) LB=LC.LB=LC. b) BE=CF.BE=CF.
geometry
P(x^2) is irreducible in Q[x]

Source: 2023 Viet Nam math olympiad for high school students D2 P7

3/25/2023
Given a polynomial with integer coefficentsP(x)=xn+an−1xn−1+...+a1x+a0,n≥1P(x)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0,n\ge 1satisfying these conditions: i) ∣a0∣|a_0| is not a perfect square. ii) P(x)P(x) is irreducible in Q[x].\mathbb{Q}[x].
Prove that: P(x2)P(x^2) is irreducible in Q[x].\mathbb{Q}[x].
algebra